GOING UP OF THE u-INVARIANT
OVER FORMALLY REAL FIELDS
CLAUS SCHUBERT
Abstract. Let F be a field of characteristic not 2, and assume F has finite
reduced stability. Let K/F be any finite extension. We prove that if the
general u-invariant u(F ) is finite, then u(K) is finite.
1. Introduction
Let F be a field of characteristic not 2. We denote by W F the Witt ring of F ,
i.e., the ring of equivalence classes of non-degenerate quadratic forms over F . Its
torsion part is denoted by Wt F . The (general) u-invariant of F is defined to be
u(F ) := max{dim ϕan | ϕ ∈ Wt F }
or u(F ) = ∞ if no such maximum exists. We are interested in the behavior of the
u-invariant under finite field extensions K/F . We say u “goes up” a field extension
K/F if finiteness of u(F ) implies finiteness of u(K), and u “goes down” if finiteness
of u(K) implies finiteness of u(F ). If F is non-formally
√ real, then u(F ) < ∞ implies
that u(K) < ∞ ([EL76, Theorem F]). If K = F ( w) is a quadratic extension
where w ∈ F × is totally positive, then u(F ) < ∞ if and only if u(K) < ∞ ([EL76,
Theorem H]). If K/F is a finite normal extension, then u(K) < ∞ implies that
u(F ) < ∞ ([Elm77, Theorem 3.2]). In particular, the finiteness of the u-invariant
goes down arbitrary quadratic extensions ([Elm77, Theorem 3.1]). However, in
general, Going Up will not hold for arbitrary quadratic extensions when the base
field is formally real. For example, the field F = R((x1 ))((x2 ))((x3 )) . . . of iterated
Laurent
series in infinitely many variables over R has u-invariant
u(F ) = 0, but
√
√
F ( −1) = C((x1 ))((x2 ))((x3 )) . . . has u-invariant u(F ( −1)) = ∞. Thus, for
Going Up, additional assumptions will be needed. We will show in this article that
assuming that F has finite reduced stability is sufficient and necessary.
In this article, a form over F means a non-degenerate quadratic form. In the
proof of the main result we use results from the theory of abstract spaces of orderings
and apply them to the theory of quadratic forms over fields. The terminology and
results on spaces of orderings can be found in [Mar96]. We collect some of the basic
definitions and results here.
Let G be an elementary 2-group, written multiplicatively, with a distinguished
element −1. A form ϕ over G is a symbol ϕ = ha1 , . . . , an i with ai ∈ G for
all i. If ϕ = ha1 , . . . , an i and ψ = hb1 , . . . , bm i are forms over G, then the sum
Date: February 1, 2007.
2000 Mathematics Subject Classification. 11E04, 11E81.
This article is based on part of the author’s Ph.D. thesis, written under the supervision of
Richard Elman.
1
2
CLAUS SCHUBERT
of ϕ and ψ is ϕ ⊕ ψ := ha1 , . . . , an , b1 , . . . , bm i. The character group of G is
χ(G) := Hom(G, {−1, 1}).
Let X be a nonempty set and G a subgroup of {−1, 1}X . Let ϕ = ha1 , . . . , an i be
a form over G. Then
dimension of ϕ is n and the signature of ϕ at α ∈ X is the
Pthe
n
integer sgnα (ϕ) := i=1 ai (α). Two forms ϕ and ψ are isometric (over X), write
ϕ∼
= ψ, if they have the same dimension and the same signature at each α ∈ X.
We say ϕ represents a ∈ G if ϕ ∼
= ha, a2 , . . . , an i with a2 , . . . , an ∈ G. The set of
all elements of G represented by ϕ is denoted by D(ϕ). A space of orderings is a
pair (X, G) satisfying the following axioms:
(1) X is a nonempty set, G is a subgroup of {−1, 1}X containing the constant
function −1, and G separates points in X (i.e., if x 6= y ∈ X then there exists g ∈ G
such that g(x) 6= g(y)).
(2) The image of the embedding X → χ(G), x 7→ (g 7→ g(x)) is closed in χ(G).
We will usually identify X with its image in χ(G) from here on.
(3) If ϕ and ψ are forms over G and x ∈ D(ϕ ⊕ ψ) then there are y ∈ D(ϕ) and
z ∈ D(ψ) such that x ∈ D(hy, zi).
In particular, the ordering space XF of a formally real field F is a space of
orderings (X, G) in the above sense with X = XF and G given by the group of
generalized square classes F × /σ(F ), where σ(F ) = {x21 + . . . + x2n | xi ∈ F × , i =
1, . . . , n} ([Mar80b, Theorem 1.3]).
The elements of X are called orderings. For an element a ∈ G we define H(a) =
{α ∈ X | α(a) = 1}, the Harrison set of a. We define a topology on X with subbasis
consisting of all Harrison sets. X is compact with this topology. A subspace of a
space of orderings is a subset Y ⊂ X closed under linear combinations of orderings,
i.e., if α1 , . . . , αn ∈ Y , then α1 α2 . . . αn ∈ Y . A subspace of a space of orderings is
itself a space
L of orderings. We say (X, G) is the direct sum of (Xi , Gi ), i ∈ I, write
(X, G) = i∈I (Xi , Gi ), if X is the disjoint union of the Xi and G consists of all
functions g ∈ {−1, 1}X satisfying g|Xi ∈ Gi , i ∈ I. The translation group of X is
gr(X) := {γ ∈ χ(G) | γX = X}. If gr(X) 6= 1, then we define the residue space
of X as (X 0 , G0 ) where G0 = gr(X)⊥ := {a ∈ G | γ(a) = 1 for all γ ∈ gr(X)} is a
subgroup of G and X 0 denotes the image of X in χ(G0 ) via restriction. In this case
(or more generally, if G0 is any subgroup of G), the space (X, G) is called a group
extension of (X 0 , G0 ). We call X an elementary indecomposable space (EI-space) if
either |X| = 1 or gr(X) 6= 1 and |X| ≥ 4.
A form ϕ over G is called isotropic if ϕ ∼
= h1, −1, a3 , . . . , an i with a3 , . . . , an ∈
G and anisotropic otherwise. Just as in the classical theory of quadratic forms
over fields, every form ϕ over G decomposes as ϕ ∼
= ih1, −1i ⊕ ϕan where ϕan is
anisotropic and i ≥ 0 an integer. Two forms ϕ and ψ are Witt equivalent if ϕan ∼
=
ψan . The equivalence classes of forms over G under this equivalence relation form a
ring called the (abstract) Witt ring of (X, G) and denoted by W (X). The cokernel
of the map W (X) → C(X, Z) sending ϕ to ϕ
b : X → Z, ϕ(α)
b
= sgnα ϕ is 2-primary
torsion (cf., for example, [Mar80b, Lemma 5.4]). Here Z is endowed with the
discrete topology. The stability st(X) of a space of orderings (X, G) is defined to
be the minimal 2-power that annihilates this cokernel, or ∞ if no such minimum
exists. If XF is the ordering space of a formally real field F , then st(XF ) is called
the reduced stability of F , denoted by str F . By [Brö74, Satz 3.17], str F ≤ n if and
only if I n+1 F = 2I n F + Itn+1 F where I n F is the n-th power of the fundamental
ideal IF in the Witt ring W F and Itn F = I n F ∩ Wt F . For convenience, we define
GOING UP OF THE u-INVARIANT
3
the reduced stability of a non-formally real field to be 0. The main structure results
we use for spaces of orderings (X, G) are summarized below.
Fact 1 ([Mar80a, Theorem 2.6 and Remark 2.7]). If st(X) < ∞ then the connected
components Xi , i ∈ I of X are EI-subspaces and W (X) consists of all the ϕ ∈
C(X, Z) satisfying ϕ|Xi ∈ W (Xi ) and sgnα ϕ ≡ sgnβ ϕ (mod 2) for all α ∈ Xi ,
β ∈ Xj , i 6= j.
Fact 2 ([Mar80a, Remark 2.7]). If (X, G) has components (Xi , Gi ), i ∈ I, and
st(X) 6= 1 is finite, then st(X) = max{st(Xi ) : i ∈ I}.
Fact 3. If st(X) < ∞ and (X, G) is a group extension of (X 0 , G0 ) then [G : G0 ] <
∞. If [G : G0 ] = 2k then st(X) = st(X 0 ) + k.
Fact 3 follows from [Mar80b, Theorem 6.4] and an easy induction argument.
2. Matching Signatures over Spaces of Orderings
Let (X, G) be a space of orderings and ϕ a form over G. We wish to find a form
ψ over G satisfying sgnα ϕ = sgnα ψ for all α ∈ X and with ψ having minimal
possible dimension. The easiest case occurs when st(X) ≤ 1. In this case, we say
(X, G) satisfies the strong approximation property (SAP). It is easily proved that
if (X, G) satisfies SAP and A and B are disjoint, closed subsets of X, then there
exists a ∈ G such that α(a) = −1 for α ∈ A and α(a) = 1 for α ∈ B (cf. [Brö74,
Satz 3.20] for the field case; the same proof works in the general case).
Lemma 1. Let (X, G) be a space of orderings satisfying SAP. Let ϕ be a form
over G with | sgnα ϕ| ≤ m for all α ∈ X. Then there exists a form ψ over G of
dimension at most m such that sgnα (ϕ − ψ) = 0 for all α ∈ X.
Proof. We induct on m. If m = 0, there is nothing to prove. For m > 0, let ϕ be a
form over G satisfying | sgnα ϕ| ≤ m for all α ∈ X and let A = {α ∈ X | sgnα ϕ ≥ 0}
and B = X\A. Note that both A and B are clopen (closed and open). Since
X satisfies SAP, there exists a ∈ G such that α(a) = −1 for all α ∈ A and
α(a) = 1 for all α ∈ B. Then | sgnα (ϕ ⊕ hai)| ≤ m − 1 for all α ∈ X, so by the
induction assumption there exists a form ψe over G such that dim ψe ≤ m − 1 and
e = 0 for all α ∈ X. Then ψ = ψe − hai works.
sgnα (ϕ ⊕ hai − ψ)
Our aim is to give a bound on the dimension of forms ψ as above where (X, G)
has finite stability. By Fact 1, all the components of X are EI-subspaces. Thus, we
will have to examine what happens to signatures of forms over a group extension
X of a space X 0 . The following lemma deals with this case.
Lemma 2. Let (X, G) be a space of orderings that is a group extension of (X 0 , G0 )
with |G/G0 | = 2, say G = G0 ∨ xG0 . Let ϕ be a form over G with | sgnα ϕ| ≤ n
for all α ∈ X. Then there are forms ϕ1 , ϕ2 over G0 such that ϕ = ϕ1 ⊕ xϕ2 and
| sgnα ϕ1 | + | sgnα ϕ2 | ≤ n for all α ∈ X. In particular, | sgnα ϕi | ≤ n for i = 1, 2
and all α ∈ X.
More generally, suppose that |G/G0 | = 2k , say G/G0 is generated by
{x1 , . . . , xk }. Let ϕ be a form over G with | sgnα ϕ| ≤ n for all α ∈ X. Then
there are forms ϕε over G0 with | sgnα ϕε | ≤ n for all α ∈ X such that
M
ϕ=
xε11 xε22 · · · xεkk ϕε
ε
4
CLAUS SCHUBERT
where the sum is taken over all possible ε = (ε1 , . . . , εk ), εi ∈ {0, 1}.
Proof. Assume that |G/G0 | = 2, say G = G0 ∨ xG0 . Let ϕ be a form over G.
Then we can write ϕ ∼
= ha1 , . . . , am , xb1 , . . . , xbk i with ai , bi ∈ G0 for all i. Let
ϕ1 = ha1 , . . . , am i and ϕ2 = hb1 , . . . , bk i. Then ϕ = ϕ1 ⊕ xϕ2 and ϕ1 , ϕ2 are forms
over G0 . Let α ∈ X be any ordering in X, and let β ∈ X be the corresponding
ordering such that β|X 0 = α|X 0 and β(x) = −α(x). Without loss of generality,
assume that α(x) = 1. Let r = sgnα ϕ1 and s = sgnα ϕ2 = sgnα xϕ2 . If r and s are
both positive or both negative, then | sgnα ϕ| = |r + s| ≤ n and so |r| + |s| ≤ n. If
r and s have opposite signs, then note that r = sgnβ ϕ1 and −s = sgnβ xϕ2 , so by
the first case, |r| + | − s| ≤ n.
Now assume |G/G0 | = 2k and G/G0 is generated by {x1 , . . . , xk }. We successively
construct subgroups Gi , i = 0, . . . , k, such that G = G0 and G0 = Gk and Gi =
Gi+1 ∨ xi+1 Gi+1 , where i = 0, . . . , k − 1. By L
using the case k = 1 above and
induction on k, it is clear that we may write ϕ = ε xε11 xε22 · · · xεkk ϕε with forms ϕε
over G0 , where the sum is taken over all possible ε = (ε1 , . . . , εk ) with εi ∈ {0, 1}.
The fact that each ϕε has signature | sgnα ϕε | ≤ n for all α ∈ X follows again from
the above case k = 1 and induction, and from the fact that for any form ϕ over G0
and any α ∈ X, we have sgnα ϕ = sgnα|X 0 ϕ.
Remark 1. The signature of a form over G is invariant under equivalence of spaces
of orderings in the following sense: if ϕ = ha1 , a2 , . . . , ak i and α ∈ X, then sgnα ϕ =
Pk
0
0
i=1 α(ai ). Now assume there is an equivalence of spaces (X , G ) ∼ (X, G), i.e.,
0
an isomorphism f : Gf
→G together with its dual isomorphism f ∗ : χ(G0 )f
→χ(G)
∗
0
such that f (X ) = X. Then for α0 ∈ X 0 , the form ϕ0 = hf (a1 ), f (a2 ), . . . , f (ak )i
has signature
0
sgnα0 ϕ =
k
X
i=1
0
α (f (ai )) =
k
X
f ∗ (α0 )(ai ) = sgnf ∗ (α0 ) ϕ.
i=1
Thus, if ψ is a form over G such that sgnα (ϕ − ψ) = 0 for all α ∈ X, then there
exists a form ψ 0 over G0 of the same dimension such that sgnα0 (ϕ0 − ψ 0 ) = 0 for all
α0 ∈ X 0 . Hence any result that we obtain on the relationship between signatures
and dimensions of forms on an ordering space (X, G) will be true for any equivalent
space (X 0 , G0 ).
We now formulate our result on signature matching of forms over ordering spaces
with finite stability. This provides the crucial part in the proof of the main result
of this article.
Theorem 1. Let (X, G) be a space of orderings of stability st X = n < ∞. Let ϕ
be a form over G with | sgnα ϕ| ≤ m for all α ∈ X. Then there exists a form ψ over
G such that dim ψ ≤ 2n−1 m (or m if n ≤ 1) and sgnα (ϕ − ψ) = 0 for all α ∈ X.
Proof. We induct on st X. If X has stability 0 or 1, then X satisfies SAP. We are
done by Lemma 1. For the induction step, we let (X, G) be a space of orderings
with st X = n > 1 and assume that the statement holds for all spaces of orderings
with stability less than n. We consider two cases.
Case 1. X is an EI-space.
As st X > 1, we have |X| > 1. Thus, by the definition of EI-space, gr(X) 6= 1
and (X, G) is a group extension of its residue space (X 0 , G0 ). By Fact 3, |G/G0 | = 2k
for some k and st X 0 = n − k. Here k > 0 since otherwise gr(X) = 1.
GOING UP OF THE u-INVARIANT
5
Let ϕ be a form over
L G with signature | sgnα ϕ| ≤ m for all α ∈ X. By Lemma
2, we may write ϕ = ε xε11 xε22 · · · xεkk ϕε where the sum is taken over all possible
ε = (ε1 , . . . , εk ) with εi ∈ {0, 1} and where ϕε are forms over G0 with signature
| sgnα ϕε | ≤ m for all α ∈ X (and hence for all α ∈ X 0 ).
Thus by the induction assumption, for each ε there is a form ψε over G0 with
dim ψε ≤ 2n−k−1 m (or m if |X 0 | =
sgnα (ϕε − ψε ) = 0 for all α ∈ X 0 , and
L1) εand
1 ε2
hence also for all α ∈ X. Let ψ = ε x1 x2 · · · xεkk ψε where again the sum is taken
over all possible ε = (ε1 , . . . , εk ) with εi ∈ {0, 1}. Then sgnα (ϕ − ψ) = 0 for all
α ∈ X and dim ψ ≤ 2k 2n−k−1 m = 2n−1 m.
Case 2. X is not an EI-space.
In this case, the connected components of (X, G) form a partition of X by
[Mar80a, Theorem 2.3]. By Fact 1, the components are EI-subspaces (Xi , Gi ) for
some
i ∈ I. The group G is identified with a subgroup of the direct product
Q
G
i . Since st(X) = n > 1, by Fact 2,
i∈I
Qwe have st(X) = max{st(Xi ) | i ∈ I}.
We denote by πi , i ∈ I, the projection of i∈I Gi on the i-th coordinate Gi .
Let ϕ = ha1 , a2 , . . . , ar i be a form over G with | sgnα ϕ| ≤ m for all α ∈ X. For
i ∈ I and j = 1, . . . , r, let aij = πi (aj ) ∈ Gi , and let ϕi = hai1 , ai2 , . . . , air i. Then
we have sgnα ϕi = sgnα ϕ for all α ∈ Xi . The spaces Xi have stability less than or
equal to n for all i ∈ I and are EI-spaces. Thus, by the induction assumption and
by case 1, there are forms ψi of dimension ki ≤ 2n−1 m over Gi such that sgnα (ϕi −
ψi ) = 0 for all α ∈ Xi , say, ψi = hbi1 , bi2 , . . . , biki i, i ∈ I. Let k = max{ki | i ∈ I}. If
i
ki < k for some i, replace ψi by the form hbi1 , bi2 , . . . biki i ⊕ k−k
2 h1Gi , −1Gi i. This
n−1
does not change the signature of ψi . So we may assume
m
Q that dim ψi = k ≤ 2
0
for i ∈ I. Let ψ = hb1 , b2 , . . . , bk i be the form over i∈I Gi with πi (bj ) = bij for
all i ∈ I and j = 1, . . . , k. Then ψ 0 ∈ W (X) by Fact 1. Pick a form ψ over G
that represents ψ 0 with dim ψ ≤ dim ψ 0 (e.g., ψ anisotropic will satisfy this). Then
sgnα (ϕ − ψ) = sgnα (ϕi − ψi ) for all α ∈ Xi and all i ∈ I, so sgnα (ϕ − ψ) = 0 for
all α ∈ X and dim ψ ≤ k ≤ 2n−1 m.
We will apply this result to the field case in the next section.
3. The um (F )-Invariant
Definition 1. Let F be a formally real field. For m ∈ N, we define
um (F ) := max dim ϕan : | sgnα ϕ| ≤ m for all α ∈ XF
ϕ∈W F
(or um (F ) = ∞ if no such maximum exists).
If F is non-formally real, we define
um (F ) := u(F ) for all m ∈ N.
We now apply Theorem 1 to the field case.
Proposition 1. Let F be a field with str F = n < ∞. Then for all m ∈ N,
um (F ) ≤ u(F ) + 2n−1 m
(or um (F ) ≤ u(F ) + m if str F = 0).
Proof. If F is non-formally real there is nothing to prove, so assume F is formally
real. Let ϕ be an anisotropic form over F with | sgnα ϕ| ≤ m for all α ∈ XF .
We may view ϕ as a form ϕ over F × /σ(F ). By Theorem 1, there exists a form
ψ = ha1 , . . . , ak i over G such that dim ψ ≤ 2n−1 m (or m if n ≤ 1) and sgnα (ϕ −
6
CLAUS SCHUBERT
ψ) = 0 for all α ∈ XF . We may view ψ as a form ψ = ha1 , . . . , ak i over F
by picking any representative ai ∈ F × of its generalized square class ai in G for
i = 1, . . . , k. By Pfister’s Local-Global Principle, the form ϕ − ψ is a torsion form,
so dim(ϕ − ψ)an ≤ u(F ). Hence
dim ϕ ≤ dim(ϕ − ψ)an + dim ψ ≤ u(F ) + 2n−1 m.
Example 1. The upper bound in Proposition 1 can be attained. For example, let
F = R((x))((y)) and ϕk = kh1, x, y, −xyi for k ≥ 1. Here n = 2 and u(F ) = 0.
For each k, the form ϕk is anisotropic and totally indefinite, has dimension 4k, and
| sgnα ϕk | ≤ 2k for all α ∈ XF . Thus, for all m = 2k, we have um (F ) ≥ 2m.
4. Going Up of the u-Invariant
In this section, we show that for√any field F with str F < ∞, the finiteness of
u(F ) implies the finiteness of u(F ( −1)) and hence the finiteness of u(K) for any
finite extension K/F . For the proof, we need the following lemma.
Lemma 3 ([EL72, Theorem 3.2]). Let F be a formally real field with ordering
space XF . Let A and B be disjoint closed subsets of X. Then there exists a positive
integer n and a form ϕ ∈ I n F such that sgnα ϕ = 0 for all α ∈ A and sgnα ϕ = 2n
for all α ∈ B. Furthermore, if F has reduced stability str F ≤ k, we may choose
n = k.
The last statement in Lemma 3 follows from [EL72, Proposition 3.7] with only
minor modifications.
Theorem 2. Let F be a field with str F = n − 1 < ∞, n ≥ 1. Then
√
1
u(F ( −1)) ≤ u(F ) + u2n −1 (F ).
2
Proof. If F is non-formally real then u2n −1 (F ) = u(F ) and the result √
follows by
[EL76, Theorem 7.1]. So we may assume F is formally real. Let K = F ( −1) and
let ψ ∈ Wt K = W K be anisotropic. Then by [Elm77, Theorem 2.1] we can write
ψ∼
= ϕK ⊥ ψ1 where ϕ is a form over F and ψ1 is a form over K with s∗ (ψ√
1 ) ∈ Wt F
anisotropic (where s∗ is the transfer induced by the F -linear functional s( −1) = 1
and s(1) = 0). Thus, dim ψ1 ≤ 21 u(F ).
For m ∈ N, let
Xm = {α ∈ XF | m2n ≤ | sgnα ϕ| < (m + 1)2n }.
Note that using the continuous function ϕ
b : XF → Z defined by ϕ(α)
b
= sgnα ϕ,
each Xm can be expressed as a finite union of clopen sets of the form ϕ
b−1 (r) for
some r ∈ Z, so each Xm is clopen. Only finitely many of them are nonempty. Let
M = {m1 , . . . , mr } consist of all the indices mi ≥ 1 such that Xmi 6= ∅. By Lemma
3, for each m ∈ M there exists ρm ∈ I n F such that | sgnα (ϕ P
− ρm )| < 2n for
r
α ∈ Xm and sgnβ (ϕ − ρm ) = sgnβ ϕ for all β ∈ XF \Xm . Let ρ = i=1 ρmi ∈ I n F .
Since str F = n − 1, we can write ρ = 2ρ1 + ρ2 with ρ1 ∈ I n−1 F and ρ2 ∈ Wt F .
Let σ = ϕ − 2ρ1 . Then
| sgnα σ| = | sgnα (ϕ − 2ρ1 )| = | sgnα (ϕ − ρ)| ≤ 2n − 1
for all α ∈ XF . Thus, dim(σK )an ≤ u2n −1 (F ), and as (2ρ1 )K = 0, we get
dim(ϕK )an = dim(σK )an .
GOING UP OF THE u-INVARIANT
7
√
Theorem 2 together with Proposition 1 establishes the following bound for u(F ( −1)):
Corollary 1. Let F be a field. If u(F ) < ∞ and str F = n < ∞ then
√
3
u(F ( −1)) ≤ u(F ) + 22n − 2n−1
2
if n > 0 and
√
3
u(F ( −1)) ≤ u(F ) + 1
2
if n = 0.
√
Remark 2. Similarly, one can find bounds for any quadratic extension K = F ( x)
of a formally real field F with finite reduced stability str F = n < ∞. If x is not
totally negative over F , then the orderings α ∈ H(x) extend to K (in particular, K
will also be formally real). If ψ ∈ Wt K, decompose ψ as in the proof of Theorem 2.
The form ψ1 has dimension ≤ 21 u(F ) and thus | sgnα ϕ| ≤ 12 u(F ) for all α ∈ H(x).
Let the form ρ1 be defined as in the proof but using sets Xm that form a cover of
H(−x) instead of XF . Define σ as ϕ − h1, −xiρ1 . Its signature | sgnα σ| is then
bounded by 21 u(F ) on H(x) and 2n+1 − 1 on H(−x), so that
√
3
u(F ( x)) ≤ u(F ) + max{22n − 2n−1 , 2n−2 u(F )}
2
√
3
if n > 0, and u(F ( x)) ≤ 2 u(F ) + max{1, 21 u(F )} if n = 0.
Combining the bound from Corollary 1 with previously known bounds one obtains bounds for finite extensions K/F :
Corollary 2. Let F be a formally real field with u(F ) < ∞ and str F = n < ∞.
Let K/F be any finite extension of F . Then
u(K) < ([K : F ] + 1)(3u(F ) + 22n+1 − 2n )
if n > 0 and u(K)
< ([K : F ] + 1)(3u(F ) + 2) if n = 0.
√
If K = K( −1) then
3
u(K) ≤ ([K : F ] + 1)( u(F ) + 22n−1 − 2n−2 ).
4
√
Proof. u(K) <√4u(K( −1)) by [EL76, Theorem
6.2]. By [Lee84, Theorem 2.10],
√
we have u(K( −1)) ≤ 21 ([K : F ] + 1)u(F ( −1)). The estimates then follow from
Corollary 1.
The following theorem summarizes the behavior of the u-invariant and reduced
stability under finite field extensions.
Theorem 3. Let F be a field. Then the following are equivalent:
(1) u(F )√
< ∞ and str F < ∞.
(2) u(F ( −1)) < ∞.
(3) u(K) < ∞ and str K < ∞ for every finite extension K/F .
(4) There exists a finite extension E/F with u(E) < ∞ and str E < ∞.
(5) u(K) < ∞ for every finite extension K/F .
8
CLAUS SCHUBERT
Proof. (1)⇒(2): This
√ is Corollary 1.
(2)⇒(1):
As
F
(
−1) is non-formally real, u(F ) < ∞ by [EL76, Theorem G].
√
If u(F ( −1)) < 2m for some m ∈ N, then by [EL76, Corollary 4.7(i)], we have
str F ≤ m.
√
(2)⇒(3):
Let K/F be any finite extension. Then u(F ( −1)) < ∞ implies that
√
u(K( −1)) < ∞ by [Elm77, Corollary 3.3(ii)], and hence u(K) < ∞ by [Elm77,
Corollary 3.3(i)]. Again by [EL76, Corollary 4.7(i)], we have str K < ∞.
(3)⇒(4), (3)⇒(5) and (5)⇒(2) are trivial.
(4)⇒(2):
If E/F is a finite extension with u(E)
< ∞ and str E < ∞ then
√
√
u(E( −1)) < ∞ by Corollary 1 and hence u(F ( −1)) < ∞ by [Elm77, Corollary
3.3(i)].
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Department of Mathematics, University of California, Los Angeles, Los Angeles,
CA 90095-1555
E-mail address: [email protected]